Mathematics - Vector Question with Solution | TestHub

Consider two vectors $\overrightarrow{\mathrm{u}}=3\hat{\mathrm{i}}-\hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{v}}=2\hat{\mathrm{i}}+\hat{\mathrm{j}}-\lambda \hat{\mathrm{k}},\lambda >0$. The angle between them is given by ${\cos }^{-1}\left(\frac{\sqrt{5}}{2\sqrt{7}}\right)$. Let $\overrightarrow{\mathrm{v}}={\overrightarrow{\mathrm{v}}}_1+{\overrightarrow{\mathrm{v}}}_2$, where ${\overrightarrow{\mathrm{v}}}_1$ is parallel to $\overrightarrow{\mathrm{u}}$ and ${\overrightarrow{\mathrm{v}}}_2$ is perpendicular to $\overrightarrow{\mathrm{u}}$. Then the value ${\left|{\overrightarrow{\mathrm{v}}}_1\right|}^2+{\left|{\overrightarrow{\mathrm{v}}}_2\right|}^2$ is equal to